A wide variety of modeling, processing and inversion algorithms require the recalculation of the response after local alterations to an initial model. For example, pre-stack finite-difference migration of seismic data provides a highly accurate means of producing images of the Earth's interior. The migration algorithm consists of recalculating the finite-difference response of small local changes to the seismic model. However, full finite-difference migration is rarely performed because of computational limitations restricting migration algorithms to the use of less accurate asymptotic techniques. Another example relates to finite-difference inversion, where recalculating the finite-difference response is the core (forward modeling step) of the algorithms.
Yet another example which is considered as being an important area of the present invention refers to so-called time-lapse seismics (or 4-D seismics). In this application it is of interest to investigate the effects that small (local) changes to the model have on the seismic response, e.g., varying water-oil-contact levels in a producing reservoir.
Also, in forward modeling, it may be of interest to re-compute the response of an altered seismic model. Forward modeling may serve as a means of learning what effects certain features of a seismic model have on the full response. Also, as the knowledge of the model evolves, or as it becomes more refined, a simulated response may need to be updated.
Another area of interest regarding the present invention lies in Amplitude Variation with Offset (AVO) calculations, where the effects of, for instance, changes of the degree of anisotropy of a cap-rock may be the target of investigation.
Furthermore, FD modeling has been used in connection with borehole measurements, simulations of tool behavior and characteristics in their operational environment. Typically, it is of interest to investigate the effects that small changes to the tool design or model parameters have on the propagation of waves in the vicinity of the tool.
The common feature of these problems is that changes to the model are often restricted to a small sub-volume, but finite-difference simulations are required for the full model with several alterations. A method that would allow full finite-difference simulations for the complete model to be corrected for these changes while only requiring calculations in the sub-volume and its neighborhood could significantly reduce the computational cost both in terms of the number of calculations and memory for storage of material parameters and variable fields.
Finite-difference methods provide an accurate way of computing seismograms from complex seismic models. However, as mentioned above, the finite-difference simulations tend to become prohibitively expensive to run on even state-of-the-art computing equipment. Therefore, different approaches have been taken to make highly accurate numerical modeling methods such as finite-difference schemes more efficient. Two major directions of effort to achieve significant computational savings can be found in the literature: (1) hybrid techniques; and (2) grid-refinement techniques.
In the U.S. Pat. No. 6,125,330, in the background art as cited therein, and in: Robertsson and Chapman, Geophysics Vol. 65, No. 3 (May-June 2000), p. 907-918, there are described methods of injecting an analytical source solution recorded during an initial simulation on a full model on a surface surrounding a smaller domain, to drive a finite-difference computation on the smaller domain. In the proposed FD-injection approach the injection surface is transparent for waves scattered by the perturbations, allowing them to “leak trough” the injection surface and to be extrapolated to a set of receivers. However, even as an improvement over other known methods, the proposed method places still high demands on the performance of absorbing boundary conditions and significantly increases the size of computational domain (and thus the computational cost).
Exact non-reflecting boundary conditions have been proposed in: Ting and Miksis, J. Acoust. Soc. Am. 80, 1825 (1986) for scattering problems based on the Kirchhoff integral. These methods use extrapolating the wavefield from an artificial surface surrounding the scatterer to the boundary of the computational domain, the exact boundary conditions are found, necessary to truncate the computational domain without generating spurious reflections. The known scheme has several advantages in that it is explicit, it only requires past values on the extrapolation surface and if an explicit difference scheme is used in the interior, the boundary condition can be solved at each time step independently. The extrapolation surface can be of arbitrary shape or size, the size of the computational domain can be of the same order as the scatterer. However the method as proposed by Ting and Miksis and as tested by others makes use of free-space Green's functions thus assuming or enforcing a homogenous background medium where the values of the extrapolated wavefield on the boundary depend only the values of the wavefield on the extrapolation surface at the retarded times t−r/c, where r/c is the traveltime of the wavefield between the extrapolation surface and the boundary.
The boundary is non-reflecting. The process neglects scattering due to an inhomogeneous medium outside the volume enclosed by the extrapolation boundary.
In Phys. Rev. Lett. 94, 164301 (2005) and in the co-owned international patent application PCT/GB2005/003852, entitled “Processing Data representing Energy propagating through a Medium” and filed on 6 Oct. 2005, the inventors of the present invention described methods using the representation theorem for the wave-equation in combination with time-reversal invariance and reciprocity, to express the Green's function between two points in the interior of the model as an integral over the response in those points due to sources regularly distributed on a surface surrounding the medium and the points. In the application reference is made to the above-cited U.S. Pat. No. 6,125,330 and it is proposed to combine the method of FD-injection of the latter with the efficient calculation of Green's calculation of the former.
In view of the above cited prior art it is an object of the invention to provide methods for improving the efficiency of seismic wave-field calculation as described in U.S. Pat. No. 6,125,330. It is a more specific object of the invention to improve the effectiveness of boundary conditions around sub-parts or domains in a larger finite-difference model.